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Sequence and series

  1. Mar 24, 2009 #1
    1. The problem statement, all variables and given/known data
    Let xn = 1/ln(n+1) for n in N.

    a) Use the definition of limit to show that lim(xn) = 0.
    b) Find a specific value of K(ε) as required in the definition of limit for each of i)ε=1/2, and ii)ε=1/10.

    3. The attempt at a solution

    a) If ε > 0 is given,
    1/ln(n+1) < ε <=> ln(n+1) > 1/ε <=> e^(ln(n+1)) > e^(1/ε) <=> n+1 > e^(1/ε)
    <=> n > e^(1/ε) - 1
    Because ε is arbitrary number, so we have n > 1/ε.
    If we choose K to be a number such that K > 1/ε, then we have 1/ln(n+1) < ε for any n > K.

    right??

    b) so.. K can be 3 for )ε=1/2, and 11 for ii)ε=1/10.
    Correct?
     
  2. jcsd
  3. Mar 25, 2009 #2

    Mark44

    Staff: Mentor

    You did fine up to the line above. You want n > e^(1/ε). That affects your answers to part b.
     
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